Energy-Efficient Train Control

Rail is potentially a very efficient form of transport, but must be convenient, reliable and cost-effective to compete with road and air transport. Optimal control can be used to find energy-efficient driving strategies for trains. This book describes the train control problem and shows how a solution was found at the University of South Australia. This research was used to develop the Metromiser system, which provides energy-efficient driving advice on suburban trains. Since then, this work has been modified to find practical driving strategies for long-haul trains. The authors describe the history of the problem, reviewing the basic mathematical analysis and relevant techniques of constrained optimisation. They outline the modelling and solution of the problem and finally explain how the fuel consumption can be minimised for a journey, showing the effect of speed limits and track gradients on the optimal driving strategy.

Section A: Introduction.- 1 The Train Control Problem.- 1.1 The Original Formulation.- 1.2 Solution of the Original Problem.- 1.3 Initial Results.- 1.4 Further Early Results.- 1.5 Metromiser.- 1.6 Long-Haul Operations.- 2 Modelling the Train Control Problem.- 2.1 The Mechanical Energy Model.- 2.2 The Fuel Consumption Model.- 2.2.1 Traction Characteristics for a Diesel-Electric Locomotive.- 2.2.2 Braking Characteristics for a Diesel-Electric Locomotive.- 2.2.3 Modelling the Control Mechanism.- 2.2.4 The Equations of Motion.- 2.3 Trains with Distributed Mass.- 3 Practical Driving Strategies.- 3.1 Approximation of Measurable Control.- 3.2 Speed-Holding.- 4 Constrained Optimisation—An Intuitive View.- 4.1 Constrained Optimisation.- 4.2 Linearisation Techniques.- 4.3 Hyperplanes and Half-Spaces.- 4.3.1 Hyperplanes in Hilbert Space.- 4.3.2 Hyperplanes in the Dual Space.- 4.3.3 Half-Spaces.- 4.4 Linear Mappings.- 4.5 The Adjoint Mapping.- 4.6 Convex Cones.- 4.6.1 Polar Cones.- 4.6.2 Positivity.- 4.6.3 Cones in the Dual Space.- 4.6.4 The Cone Separation Theorem.- 4.6.5 The Cone Inclusion Theorem.- 4.6.6 Farkas Theorem for Convex Cones.- 4.7 The Optimisation Theorem.- 4.8 Linear Constrained Minimisation.- 4.9 The Kuhn-Tucker Conditions.- 4.10 The Pontryagin Principle.- Section B: Analysis of the Mechanical Energy Model.- 5 Existence of an Optimal Strategy.- 5.1 Introduction.- 5.2 Precise Formulation of the Train Control Problem.- 5.3 Sufficient Conditions for a Feasible Strategy.- 5.4 Existence of an Optimal Strategy.- 5.5 Conclusions.- 5.6 Appendix—Convergence in Banach Spaces.- 6 Necessary Conditions for an Optimal Strategy.- 6.1 An Equivalent Formulation of the Train Control Problem.- 6.2 Necessary Conditions for an Optimal Strategy.- 6.3 The Adjoint Differential Equation and the Pontryagin Principle.- 6.4 Conclusions.- 6.5 Appendix—The Radon Integral.- 7 Determination of Optimal Driving Strategies.- 7.1 A Special Case of the Train Control Problem.- 7.2 The Nature of the Optimal Strategy.- 7.3 The Complete Solution.- 7.4 Examples.- 7.5 Summary.- 7.6 Limitations of the Model.- Section C: Analysis of the Fuel Consumption Model.- 8 Critical Speeds and Strategies of Optimal Type.- 8.1 Introduction.- 8.1.1 A Driver Perspective on the Problem.- 8.1.2 A Well-Posed Problem.- 8.2 Formulation of the Train Control Problem.- 8.2.1 The Control Strategies.- 8.2.2 The Equations of Motion.- 8.2.3 A Precise Statement of the Problem.- 8.3 The Nature of the Resistive Acceleration.- 8.4 The Fundamental Speed Profiles.- 8.5 Necessary Conditions for a Strategy of Optimal Type.- 8.6 Calculating a Feasible Strategy of Optimal Type.- 8.7 Numerical Examples.- 8.8 Conclusions and Future Developments.- 9 Minimisation of Fuel Consumption.- 9.1 Additional Notation.- 9.2 Approximating the Minimum-Cost Strategy.- 9.3 Approximate Speed-Holding Strategies.- 9.4 The Structure of an Optimal Strategy.- 9.4.1 The Power Phase.- 9.4.2 The Transition from Power to Speed-Hold.- 9.4.3 The Transition from Speed-Hold to Coast.- 9.4.4 The Coast Phase.- 9.5 A Speed-Holding Strategy of Optimal Type.- 10 A More General Model.- 10.1 The Equations of Motion.- 10.2 Statement of the Train Control Problem.- 10.3 The Nature of the Model.- 10.4 The Main Results.- 10.5 The Fundamental Speed Profiles.- 10.6 Necessary Conditions for a Strategy of Optimal Type.- 10.7 The Critical Speeds.- 10.8 Examples.- 10.9 Approximating the Minimum-Cost Strategy.- 10.10 Approximate Speed-Holding Strategies.- 10.11 The Structure of an Optimal Strategy.- 10.11.1 The Power Phase.- 10.11.2 The Transition from Power to Speed-Hold.- 10.11.3 The Transition from Speed-Hold to Coast.- 10.11.4 The Coast Phase.- 10.12 A Speed-Holding Strategy of Optimal Type.- 11 Speed Limits.- 11.1 Vehicle Model.- 11.2 Journey Model.- 11.3 Fundamental Speed Profiles.- 11.4 Strategies of Optimal Type.- 11.5 Critical Speeds.- 11.6 Examples.- 12 Non-Constant Gradient.- 12.1 Notation.- 12.2 Solution of the Equations of Motion.- 12.3 Track Gradient Analysis and Terminology.- 12.4 The Speed Profiles.- 12.5 The Constraints.- 12.6 Necessary Conditions for a Strategy of Optimal Type.- 12.6.1 The Lagrangean Function.- 12.6.2 The Key Equations.- 12.6.3 The Kuhn-Tucker Conditions.- 12.7 Derivation of the Key Equations.- 12.7.1 Calculation of some Useful Derivatives.- 12.7.2 The Kuhn-Tucker Equations.- 12.8 An Alternative Form for the Key Equations.- 12.9 The Strategies of Optimal Type.- 12.9.1 The Properties of the Effective Energy Density Function.- 12.9.2 The Structure of a Strategy of Optimal Type.- 12.10 An Algorithm for Solving the Key Equations.- 12.11 Examples for Non-Steep Track.- 12.11.1 Level Track Strategies Applied to Track with Small Gradients.- 12.11.2 Strategies of Optimal Type on Track with Small Gradients.- 12.12 Examples for Steep Track.- 13 Continuously Varying Gradient.- 13.1 Some Additional Notation.- 13.2 A General Form for the Key Equations.- 13.3 Necessary Conditions for a Strategy of Optimal Type.- 13.3.1 An Intuitive Derivation of the Key Equations.- 13.3.2 Lagrangean Function and Kuhn-Tucker Equations.- 13.3.3 Some Results from Perturbation Theory.- 13.3.4 Calculation of some Useful Derivatives.- 13.4 An Algorithm for Solving the Key Equations.- 14 Practical Strategy Optimisation.- 14.1 A Simple Journey.- 14.2 Undulating Track.- 14.3 Speed-Holding on Steep Track.- 14.4 Overlapping Control Intervals.- 14.4.1 Overlapping Control Intervals for Steep Sections.- 14.4.2 Other Overlaps.- 14.5 Initial and Final Speeds.- 14.6 Speed Limits.- 14.7 A Practical Algorithm for Energy-Efficient Strategies.- 14.7.1 Overview.- 14.7.2 Speed-Holding.- 14.7.3 Speed-Holding with Speed Limits.- 14.7.4 Initial and Final Phases.- 14.7.5 Calculating Journey Time.- 14.7.6 Finding the Correct Hold Speed.- References.